Derivative of y=x^4cos2x

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x^{4}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

   $g{\left(x \right)} = \cos{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x$.

   2. The derivative of cosine is negative sine:

      $$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $2$

      The result of the chain rule is:

      $$- 2 \sin{\left(2 x \right)}$$

   The result is: $- 2 x^{4} \sin{\left(2 x \right)} + 4 x^{3} \cos{\left(2 x \right)}$

2. Now simplify:

   $$2 x^{3} \left(- x \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right)$$

The answer is:

$$2 x^{3} \left(- x \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right)$$